Source: https://guthlabs.ai/docs/instruments/belt-friction-capstan-equation/mechanism/

instrument mechanism / verified-clean

# Belt Friction (Capstan Equation) mechanism

The inspectable method record for Belt Friction (Capstan Equation), kept separate from implementation and qualification.

Status: **VERIFIED_CLEAN**Lifecycle: **verified-clean**

## Model scope

The core capstan limiting relation is standard, but the specification needs an explicit impending-slip/maximum-ratio label and an exact definition of the alternative "max holding load" output. The empty honest_label is a concern because the equality is a limiting condition, not a universal operating tension ratio.

- T1/T2 = e^(μθ) is the limiting capstan relation at impending slip, not the general tension ratio during static no-slip operation; generally T1/T2 ≤ e^(μθ).

- The claim that the instrument outputs a "max holding load" is underspecified: the loaded side, applied load, anchor condition, and whether T1 or T2 is solved must be defined.

- The formula requires positive tension values and a nonnegative friction coefficient; these input constraints are not stated.

## Units and conventions

- θ must be supplied in radians; degrees must be converted using θ_rad = θ_deg × π/180.

- μ is dimensionless and is assumed to be a constant friction coefficient over the contact surface.

- T1 and T2 use the same force units, such as newtons; the ratio is dimensionless.

- The equality assumes a flexible belt or rope, uniform contact, negligible bending stiffness and centrifugal effects, and impending slip under a Coulomb-friction model.

## Inspectable vector

One of 4 source-bound review vectors:

````
{
  "inputs": {
    "T2_N": 500,
    "mu": 0.2,
    "theta_rad": 1.5707963267948966
  },
  "expected_outputs": {
    "T1_N": 684.5535
  },
  "working": "μθ = 0.2 × 1.5707963268 = 0.3141592654. e^(μθ) = e^0.3141592654 ≈ 1.369107. T1 = T2 × e^(μθ) = 500 × 1.369107 = 684.5535 N."
}
````

## Failure boundary

- If T2 = 0, the stated ratio T1/T2 is undefined, even though direct evaluation of T1 = T2e^(μθ) gives zero.

- A degree value passed as radians produces an incorrect result; the interface must label the angle unit explicitly.

- Negative θ, negative μ, or negative tensions are physically invalid or reverse the intended tight/slack-side interpretation and should be rejected.

- For very large μθ, e^(μθ) can overflow numerically; an upper input bound or logarithmic handling may be needed.

- The equality should not be presented as an achievable ordinary operating ratio unless impending slip is explicitly specified.

## Other records

[01**Overview**](<https://guthlabs.ai/docs/instruments/belt-friction-capstan-equation/>)[03**Implementation**](<https://guthlabs.ai/docs/instruments/belt-friction-capstan-equation/implementation/>)[04**Qualification**](<https://guthlabs.ai/docs/instruments/belt-friction-capstan-equation/qualification/>)

## Sources and evidence

These pointers bind the page to evidence. They do not grant authority or collapse verification into deployment.

- `VERIFIED_LEDGER.json`

- `evidence-tranche4/tranche4-results.json`